Harmonic Measure - The Harmonic Measure of A Diffusion

The Harmonic Measure of A Diffusion

Consider an Rn-valued Itō diffusion X starting at some point x in the interior of a domain D, with law Px. Suppose that one wishes to know the distribution of the points at which X exits D. For example, canonical Brownian motion B on the real line starting at 0 exits the interval (−1, +1) at −1 with probability ½ and at +1 with probability ½, so B_{τ(−1, +1)} is uniformly distributed on the set {−1, +1}.

In general, if G is compactly embedded within Rn, then the harmonic measure (or hitting distribution) of X on the boundary ∂G of G is the measure μ_Gx defined by

for x ∈ G and F ⊆ ∂G.

Returning to the earlier example of Brownian motion, one can show that if B is a Brownian motion in Rn starting at x ∈ Rn and D ⊂ Rn is an open ball centred on x, then the harmonic measure of B on ∂D is invariant under all rotations of D about x and coincides with the normalized surface measure on ∂D